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Edge-Disjoint Hamiltonian Cycles in De Bruijn Graphs
The purpose of this thesis is to examine the number of edge-disjoint Hamiltonian cycles in de Bruijn graphs using ideas from finite field theory, particularly linear recurring sequences. It is known that the de Bruijn graph B(d,n) admits d-1 disjoint Hamiltonian cycles when d is a power of 2, and it is conjectured that all de Bruijn graphs B(d,n) admit d-1 disjoint Hamiltonian cycles. The conjecture also states that for every de Bruijn graph there exists a Hamiltonian cycle to which a particular function, defined in chapter 4, can be applied to obtain d-2 additional Hamiltonian cycles. I have shown for several specific de Bruijn graphs that this method does not work on Hamiltonian cycles obtained using linear recurring sequences.Keywords: de Bruijn graphs, Hamiltonian cycles, interconnection networks, linear recurring sequence
Hamilton cycles, minimum degree and bipartite holes
We present a tight extremal threshold for the existence of Hamilton cycles in
graphs with large minimum degree and without a large ``bipartite hole`` (two
disjoint sets of vertices with no edges between them). This result extends
Dirac's classical theorem, and is related to a theorem of Chv\'atal and
Erd\H{o}s.
In detail, an -bipartite-hole in a graph consists of two disjoint
sets of vertices and with and such that there are no
edges between and ; and is the maximum integer
such that contains an -bipartite-hole for every pair of
non-negative integers and with . Our central theorem is that
a graph with at least vertices is Hamiltonian if its minimum degree is
at least .
From the proof we obtain a polynomial time algorithm that either finds a
Hamilton cycle or a large bipartite hole. The theorem also yields a condition
for the existence of edge-disjoint Hamilton cycles. We see that for dense
random graphs , the probability of failing to contain many
edge-disjoint Hamilton cycles is . Finally, we discuss
the complexity of calculating and approximating
Towards obtaining a 3-Decomposition from a perfect Matching
A decomposition of a graph is a set of subgraphs whose edges partition those
of . The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011
states that every connected cubic graph can be decomposed into a spanning tree,
a 2-regular subgraph, and a matching. It has been settled for special classes
of graphs, one of the first results being for Hamiltonian graphs. In the past
two years several new results have been obtained, adding the classes of plane,
claw-free, and 3-connected tree-width 3 graphs to the list.
In this paper, we regard a natural extension of Hamiltonian graphs: removing
a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely,
removing a perfect matching from a cubic graph leaves a disjoint union
of cycles. Contracting these cycles yields a new graph . The graph is
star-like if is a star for some perfect matching , making Hamiltonian
graphs star-like. We extend the technique used to prove that Hamiltonian graphs
satisfy the 3-decomposition conjecture to show that 3-connected star-like
graphs satisfy it as well.Comment: 21 pages, 7 figure
A Distributed algorithm to find Hamiltonian cycles in Gnp random graphs
In this paper, we present a distributed algorithm to find Hamiltonian cycles in random binomial graphs Gnp. The algorithm works on a synchronous distributed setting by first creating a small cycle, then covering almost all vertices in the graph with several disjoint paths, and finally patching these paths and the uncovered vertices to the cycle. Our analysis shows that, with high probability, our algorithm is able to find a Hamiltonian cycle in Gnp when p_n=omega(sqrt{log n}/n^{1/4}). Moreover, we conduct an average case complexity analysis that shows that our algorithm terminates in expected sub-linear time, namely in O(n^{3/4+epsilon}) pulses.Postprint (published version
Routing and broadcasting in two-dimensional linear congruential graphs of degree four
"In this thesis, we consider the problems of routing and broadcasting in 2-D LC graphs of degree 4 in which f1 generates a Hamiltonian cycle, and f2 generates a few disjoint cycles.
2-factors with k cycles in Hamiltonian graphs
A well known generalisation of Dirac's theorem states that if a graph on
vertices has minimum degree at least then contains a
-factor consisting of exactly cycles. This is easily seen to be tight in
terms of the bound on the minimum degree. However, if one assumes in addition
that is Hamiltonian it has been conjectured that the bound on the minimum
degree may be relaxed. This was indeed shown to be true by S\'ark\"ozy. In
subsequent papers, the minimum degree bound has been improved, most recently to
by DeBiasio, Ferrara, and Morris. On the other hand no
lower bounds close to this are known, and all papers on this topic ask whether
the minimum degree needs to be linear. We answer this question, by showing that
the required minimum degree for large Hamiltonian graphs to have a -factor
consisting of a fixed number of cycles is sublinear in Comment: 13 pages, 6 picture
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
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